Sequences and Series

Maggie Schildt

7 Sequences and Series

Sequences and series are very much intertwined with one another. Sequences alone are one concept and series are based off the idea of summing the values of a sequence in order to construct a series. We will break these two concepts into their own sections and address each individually.

7.1 Sequences

We begin this section by discussing sequences.

Definition.

A sequence $(x_{n})$ of real numbers is a function $f:\mathbb{N} \rightarrow \mathbb{R}$ , where $x_{n}=f(n)$ .

In short a sequence“is a function whose domain is a set of consecutive natural numbers beginning with 1″[19]. Some sequences are a set number of values long, called finite sequences, where as others are infinitely long, we call these infinite sequences. When we consider what a sequence looks like we imagine an expression representing the general term for the sequence. This general term is usually represented by $a_{n}$ . Observe the following finite sequence example.

Example:

Let $a_{n} = \frac{5}{6^{n-1}}$ find the first 3 terms of the sequence.

$a_{1} = \frac{5}{6^{1-1}} = \frac{5}{6^{0}}=5$

$a_{2} = \frac{5}{6^{2-1}} = \frac{5}{6^{1}}=\frac{5}{6}$

$a_{3} = \frac{5}{6^{3-1}}=\frac{5}{6^{2}}=\frac{5}{36}$

Now there are some sequences that utilize the value of the last term to calculate the value of the next. We call this a recurrence relation. One example of this type of sequence is the Fibonacci sequence where $F_{n} = F_{n-2} + F_{n-1}$ . We see in this sequence that the two previous terms are used to calculate the value of the next term. We can observe sequences to find trends and from this we can conclude if a sequence converges or diverges. This will allow us to dictate if a sequence is reaching a quantifiable limit or not. We will define convergence and divergence later on in this section.

7.2 Series

Moving on from sequences to series. We define a series as a formal expression for the sum of the terms of a sequence. We usually denote this as

$\begin{equation*} \sum_{n=1}^{\infty}{a_{n}} \end{equation*}$

When we expand this notation we get the following calculation:

$\begin{equation*} a_{1}+a_{2}+...+a_{n}+... \end{equation*}$

There are various general series we come across in calculus [19]:

Arithmetic Series

$\begin{equation*} \sum_{k=1}^{n}{k} = 1+2+3+...+n = \frac{n(n+1)}{2}. \end{equation*}$

This series is very straight forward, we have the sum of values starting from $k=1$ with the following value increasing by 1. Now we may have a manipulated version of this series which converts every other value to a negative. This type of series is categorized as an alternating series but it should be noted that an arithmetic series will not always directly follow this pattern and can easily be altered in various ways. Much like our other section under calculus we are given the structure of basic formulas, with the knowledge that these series will have alterations applied which will effect the results.

Harmonic Series

$\begin{equation*} \sum_{k=1}^{n}{\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}}\approx ln(n). \end{equation*}$

As we can see by observing the trend of the harmonic series there is an increase in the denominator such that the values being added to the sum total are getting smaller and smaller. Now of course there are many version of the harmonic series. We can change the value of he numerator, or we can place $k$ to a power of some natural number.

Geometric Series

$\begin{equation*} \sum_{k=0}^{n}{ar^{k}}=a+ar+ar^{2}+...+ar^{n}=a\cdot\frac{r^{n+1}-1}{r-1}. \end{equation*}$

With the geometric series we are considering the terms to be a constant $a$ multiplied by the preceding term $r$ . In a general form the series appears as

$\begin{equation*} a+ar+ar^{2}+... \end{equation*}$

If the value for $r$ is greater than one then the series will trend to infinity in which we conclude that the series diverges. We will discuss divergence and convergence shortly. If $r$ is less than negative 1 then the series will appear to alternate between positive values and negative values and again we say the series diverges. Only when $-1<r<1$ does a geometric series converge [19]. There is a different notation for this situation given as

$\begin{equation*} S_{n}=\frac{a}{1-r} \end{equation*}$

Example:

Let $r=\frac{1}{2}$ we have the following geometric series

$\begin{equation*} \sum_{k=0}^{\infty}{\frac{1}{2^{k}}}=1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n}}=\frac{1}{1-\frac{1}{2}}=2 \end{equation*}$

The Power Series

With the example for geometric series in mind we introduce the power series [19]. A power series is any series taking on the form

$\begin{equation*} \sum_{n=0}^{\infty}{a_{n}x^{n}} \end{equation*}$

We see it is called the power series because of the expression containing a higher power variable [19]. Now not every power series needs to be centered around 0 in which case they are centered around some $a$ denoted as

$\begin{equation*} \sum_{k=0}^{\infty}{a_{k}(x-a)^{k}} \end{equation*}$

What we mean when discussing a series being centered around $0$ or $a$ is that every series has an interval. We assume the interval is centered at $0$ if it is not specified otherwise. When a series is centered at $a$ then we need to account for this within the formula for the series which is why we include the vital piece $(x-a)$ in the expression otherwise the series will act like it is centered around $0$ and your sum will be incorrect for the interval centered at $a$ .

7.3 Convergence and Divergence

Before we can go any further we need to discuss divergence and convergence. It has already been brought up in our discussion of limits of a sequence and in our discussion on geometric series but we will explain this further. Divergence and convergence of a series help to define the trend of a series. We calculate convergence and divergence of a sequence through partial sums. Partial sums are the sums of sections of series, observe the following example.

Example:

Consider the series

$\begin{equation*} \sum_{k=1}^{n}{\frac{1}{k^{3}}=1+\frac{1}{8}+\frac{1}{27}+...+\frac{1}{n^{3}}} \end{equation*}$

The first three partial sums are as follows

$\begin{equation*} \begin{split} S_{1}&=1\\ S_{2}&=1+\frac{1}{8}\\ S_{3}&=1+\frac{1}{8}+\frac{1}{27} \end{split} \end{equation*}$

Here we see that we are taking small portions of the values for the series and summing them up as partial sums to the series. With this in mind we define a series to be convergent if the limit for the sum of the series approaches a quantifiable limit. For some finite series this simply requires finding the total sum of the series. In order to quantify the convergence of an infinite series we use the limit convergence of the partial sums to determine if an infinite series converges. Conversely, a series diverges if limit cannot be calculated. Thus we can say any series or sequence that does not converge, diverges. Consider the two examples below.

Example:

Consider the series

$\begin{equation*} \sum_{k=0}^{n}{2^{k}}=1+2+4+8+...+2^{n} \end{equation*}$

To determine the convergence or divergence of the series we take the limit as $n$ approaches $\infty$ .

$\begin{equation*} \lim_{n\rightarrow \infty}{2^{n}}={\infty} \end{equation*}$

As $n$ approaches $\infty$ we see that $f(x) = 2^{n}$ grows without bound and thus we can conclude that the series diverges.

This is an example of a series which diverges which ultimately means no limit can be reached because the series will continue to grow infinitely larger as $n$ approaches infinity. Consider now an example of a convergent series.

Example:

Consider the series

$\begin{equation*} \sum_{k=0}^{n}{\frac{x^{2}}{x^{2}+10}} \end{equation*}$

To find if the series converges or diverges we take the limit as $n$ approaches infinity.

$\begin{equation*} \lim_{n\rightarrow \infty}{\frac{x^{2}}{x^{2}+10}}= \lim_{n\rightarrow\infty}{\frac{\frac{x^{2}}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{10}{x^{2}}}}=\lim_{n\rightarrow\infty}{\frac{1}{1+\frac{10}{x^{2}}}} \end{equation*}$

We see that as $n$ approaches $\infty$ , $\frac{10}{x^{2}}$ approaches 0 since the numerator stays constant while the denominator grows without bound. Thus we can calculate the limit to be as follows

$\begin{equation*} \lim_{n\rightarrow\infty}{\frac{1}{1+\frac{10}{x^{2}}}}=\lim_{n\rightarrow\infty}{\frac{1}{1+0}}=1 \end{equation*}$

Thus we can conclude that the series converges to 1 given that the limit as $n$ approaches infinity is 1.

7.4 Series Tests

Now that we have covered convergence and divergence of a series we can briefly cover the different tests for convergence or divergence. It should be noted that these tests exist because not all series are as easy to test for as the examples given in the last section.

Integral Test

Let $f(x)$ be a continuous, positive, and decreasing function on the interval from 1 to infinity [8]. Let us define $f(n)=a_{n}$ for a series then

$\begin{equation*} $\sum_{n=1}^{\infty}{a_{n}}$ \ \text{converges} \ $\Longleftrightarrow\int_{1}^{\infty}{f(x) dx}$ \ \text{converges}. \end{equation*}$

What this means is that if the integral for $f(x)$ from $1$ to infinity converges then the series for $a_{n}$ on the same interval, converges and vice verse. We apply this to an example.

Example:

Let $\sum_{n=1}^{\infty}{\frac{1}{n^{2}}}$ be our series such that $a_{n}=\frac{1}{n^{2}} = f(n)$ . We can note here that $f(x)$ is positive, continuous on the interval $[1, \inf)$ and decreasing. Thus we apply the integral test to test for convergence.

$\begin{equation*} \int_{1}^{\infty}{\frac{1}{x^{2}} dx}=-\frac{1}{x}|_{1}^{\infty} \end{equation*}$

Now we cannot just substitute infinity for $x$ and solve we need to take the limit as $n$ approaches infinity for $-\frac{1}{x}$ and solve for the limit.

$\begin{equation*} \lim_{n\rightarrow\infty}{\frac{1}{x}}=0 \end{equation*}$

Solving for the limit we see as $n$ approaches infinity, $\frac{1}{x}$ approaches $0$ . Thus we say the limit approaches $0$ giving us the following result for the integral.

$\begin{equation*} \int_{1}^{\infty}{\frac{1}{x^{2}} dx}=0+1=1 \end{equation*}$

Thus we conclude that the integral converges and thus the series $\sum_{n=1}^{\infty}{\frac{1}{n^{2}}}$ also converges by the integral test.

The Comparison Test

If we have two series, $a_{n}$ and $b_{n}$ where $a_{n}, b_{n}>0$ for all values of $n$ then by the comparison test we have the following rules [8].

Let $a_{n}\leq b_{n}$ , if $\sum_{n=1}^{\infty}{b_{n}}$ converges, then $\sum_{n=1}^{\infty}{a_{n}}$ converges.
Let $a_{n}\geq b_{n}$ , if $\sum_{n=1}^{\infty}{b_{n}}$ diverges, then $\sum_{n=1}^{\infty}{a_{n}}$ diverges.

In the first rule we see that if the expression $b_n$ is greater than $a_{n}$ and the series for $b_{n}$ converges, then all series smaller than $b_{n}$ , including $a_{n}$ also converge. And following this, the second rule tells us that if a smaller expression $b_{n}$ diverges then a greater expression $a_{n}$ must also diverge.

The Limit Comparison Test

The limit comparison test uses the same concept of comparing convergence or divergence between two series. For the limit comparison tests we again let $a_{n}, b_{n} >0$ for all values of $n$ . Then the following rules are given [8].

If $\lim_{n\rightarrow\infty}{\frac{a_{n}}{b_{n}}}>0$ , then $\sum_{n=1}^{\infty}{a_{n}}$ converges $\Longleftrightarrow$ $\sum_{n=1}^{\infty}{b_{n}}$ converges.
If $\lim_{n\rightarrow\infty}{\frac{a_{n}}{b_{n}}} = 0$ and $\sum_{n=1}^{\infty}{b_{n}}$ converges, then $\sum_{n=1}^{\infty}{a_{n}}$ converges.
If $\lim_{n\rightarrow\infty}{\frac{a_{n}}{b_{n}}}=\infty$ and $\sum_{n=1}^{\infty}{b_{n}}$ diverges, then $\sum_{n=1}^{\infty}{a_{n}}$ diverges.

What these rules equate to is very similar to the comparison tests, depending on the value of $b_{n}$ being greater or less than $a_{n}$ and divergent or convergent then $a_{n}$ will also be divergent or convergent respectively.

The Alternating Series Test

The alternating series test involves series which alternate between positive and negative values. Consider the example $\sum_{n=1}^{\infty}{(-1)^{n}n^2}$ . The series would look like the following

$\begin{equation*} S_{n}=(-1)+(4)+(-9)+(16)+...+(-1)^{n}n^{2} \end{equation*}$

We can see in this example that the series alternates from positive values to negative values depending on $n$ . In order to test for convergence of a series of this kind we are given the following rule [8].

Let $a_{n} = (-1)^{n+1}b_{n}, b_{n}>0$ for all values of $n$ . If $b_{n}$ is decreasing then

$\begin{equation*} \text{If} \ \lim_{n\rightarrow\infty}{b_{n}}=0 \ \text{then} \ \sum_{n=1}^{\infty}{a_{n}} \ \text{converges}. \end{equation*}$

Consider the following example.

Example:

Consider the series $\sum_{n=1}^{\infty}{\frac{(-1)^{n}n}{5n^{2}+1}}$ . Let $b_{n} = \frac{n}{5n^{2}+1}$ , if we take the limit of $b_{n}$ we get the following.

$\begin{equation*} \lim_{n\rightarrow\infty}{\frac{n}{5n^{2}+1}}=\lim_{n\rightarrow\infty}{\frac{\frac{n}{n^{2}}}{\frac{5n^{2}}{n^{2}}}+\frac{1}{n^{2}}}=\lim_{n\rightarrow\infty}{\frac{\frac{1}{n}}{5+\frac{1}{n^{2}}}}=0 \end{equation*}$

Thus we can conclude that $\lim_{n\rightarrow\infty}{b_{n}}=0$ and we can conclude that the series

$\begin{equation*} \sum_{n=1}^{\infty}{\frac{(-1)^{n}n}{5n^{2}+1}} \end{equation*}$

Converges by the alternating series test.

The alternating series test can also be proven absolutely through the next test we are going to discuss.

The Absolute Convergence Test

This next test is quite straight forward and relates to the alternating series test in the sense that instead of finding the limit for a related series we simply take the absolute value of the series to test for convergence. If $\sum_{n=1}^{\infty}{|a_{n}|}$ converges, then $\sum_{n=1}^{\infty}{a_{n}}$ converges [8].

If we consider the alternating series test in which $a_{n}$ is defined as $(-1)^{n+1}b_{n}$ and we take the absolute value for $a_{n}$ we get

$\begin{equation*} |a_{n}|=|(-1)^{n+1}b_{n}|=(1^{n+1})b_{n}=b_{n} \end{equation*}$

There are more technical comparison tests used to solve complex series problems but for now we will only focus on the tests provided above. Moving onto the next section we will delve deeper into differential equations than what has been provided in the derivatives section of the chapter.

7 Sequences and Series

License

Share This Book